A ray of light is incident from a denser to a rarer medium. The critical angle for total internal reflection is \({\theta _{iC}}\) and Brewster's angle of incidence is \({\theta _{iB}}\), such that \(\sin \,{\theta _{iC}}/\sin \,{\theta _{iB}} = \eta  = 1.28\). The  relative refractive index of the two media is

The ratio of de-Broglie wavelength of an \(\alpha\)-particle and a proton accelerated from rest by the same potential is \(\frac{1}{\sqrt{ m }}\), the value of \(m\) is \(........\)

Four identical discs each of mass ' \(M\) ' and diameter ' \(a\) ' are arranged in a small plane as shown in figure. If the moment of inertia of the system about \(OO ^{\prime}\) is \(\frac{ x }{4} Ma ^{2}\). Then, the value of \(x\) will be \(......\)

A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample is _______.

\(n\) mole a perfect gas undergoes a cyclic process \(ABCA\) (see figure) consisting of the following processes. \(A \rightarrow B :\) Isothermal expansion at temperature \(T\) so that the volume is doubled from \(V _{1}\) to \(V _{2}=2 V _{1}\) and pressure changes from \(P _{1}\) to \(P _{2}\) \(B \rightarrow C :\) Isobaric compression at pressure \(P _{2}\) to initial volume \(V _{1}\) \(C \rightarrow A\) : Isochoric change leading to change of pressure from \(P _{2}\) to \(P _{1}\) Total workdone in the complete cycle \(ABCA\) is

Number of molecules in a volume of \(4\, cm ^{3}\) of a perfect monoatomic gas at some temperature \(T\) and at a pressure of \(2\, cm\) of mercury is close to \(?\) (Given, mean kinetic energy of a molecule (at \(T\) ) is \(4 \times 10^{-14}\) erg, \(g=980\, cm / s ^{2}\), density of mercury \(=13.6\, g / cm ^{3}\))

Match List - I with List - II.

List - I	List - II
A. Boltzmann constant	I. \([M^{-1}L^3T^{-2}]\)
B. Stefan's constant	II. \([ML^2T^{-1}]\)
C. Planck's constant	III. \([ML^2T^{-2}K^{-1}]\)
D. Gravitational constant	IV. \([ML^0T^{-3}K^{-4}]\)

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A point light source emits E.M. waves in free space. A detector, placed at a distance of \(L\) m, measures the intensity as \(I_o\). The detector is now shifted to another location on the same spherical surface ensuring the angle between original location and new location as \(45°\). The measured intensity at new location will be _______.

Match the List\(-I\) with List\(-II\).

List \(I\)	List \(II\)
\(A\) \(AC\) generator	\(I\) Presence of both \(L\) and \(C\)
\(B\) Transformer	\(II\) Electromagnetic Induction
\(C\)Resonance phenomenon to occur	\(III\) Quality factor
\(D\)Sharpness of resonance	\(IV\) Mutual Inductance

Choose the correct answer from the options given below: